Two World-theories (neither one especially stringy)

[marcus]

Imagine 6 points, 3 in the present and 3 in the future------from here on we only are dealing with 2 layers at a time so I will choose them to be present and future and not refer to past.

So there are two triangles, call the downstairs triangle 134 and the upstairs triangle 256.

Now in this little venue marked by these 6 points, we can imagine a tet named 3456------this is not a spatial tet, it is timelike! It goes slanting up between layers,

and by joining this tet to apexes 1 and 2, we can make two chunks!

HERE IS WHAT MOVE (2,4) DOES.

It erases tet 3456, and shoots a line from 1 to 2, and the same bulk (which used to be TWO chunks butted together at a shared tet) is now divided up into FOUR chunks all meeting at the common line 12.

The authors write this move like this, where the underline shows what the chunks have in common

13456 + 23456 goes to 12345 +12346 + 12356 + 12456

-----analogy in 3D----
this is analogous to a move we described earlier that you can do entirely in 3D where you have two tets meeting at a triangle and you erase the triangle and shoot a line from the east vertex to the west vertex and then you have three tets meeting at that line.
---------

Now let's see what is left to do. We have dealt with at least one instance of the move (2,4)

-----comment-----
keep remembering the analogy of shuffling a deck of cards.
if you do enough shuffles you will explore all the possible orderings of the deck.
these moves are very simple modifications of a 4D triangulated geometry ("all the geometry is in the gluing") and they are only dealing with one small local cluster of chunks chosen randomly from perhaps zillions.
these moves are like a shuffle so simple that it only swaps two cards or permutes 3 or 4 cards.
but if you do enough of these very simple shuffles then in the end you make a kind of random walk thru the whole space of possibilities.

this is at the core of the MonteCarlo method which the authors have programmed, which explores the 4D geometries of their small universes
(so far at most a third of a million chunks) evolving under the Einstein rules of dynamic geometry------which rules Tullio Regge translated into rules about simplexes.

you might like to check out the animations at Jan Ambjorn website.
they give some of the flavor.

 

[marcus]

there is a minor variation on the move (2,4) just described
I mentioned that the set up was 3 points in the present and 3 in the future and it can also be set up with 4 and 2-----for example 4 points in the present and 2 in the future.

the move is written exactly the same way

13456 + 23456 goes to 12345 +12346 + 12356 + 12456

and the same thing happens as described in the previous post.

 

[godzilla7]

hi guys not sure if this is appropriate to this discussion, but seeing as you mentioned the big bang; read an article about the IMAP data being flawed because of polorizations in our solar system, if this is true does it have implications on the amount of matter we have calculated, and if it does would dark matter be irrelevant, kind of a naive question but the article was a little vague can someone clarify the errors and make a guess at the implications

 

[marcus]

... read an article about the IMAP data being flawed because of polorizations in our solar system, if this is true does it have implications ...

I read an article about that too. I think you mean WMAP.
the first few bumps in the CMB seem "too" alligned with the solar system for that to have been an accident. It could be coincidence, or it could indicate that something we don't know about in the the solar system is acting as a source or sink of microwave and affecting the low-order poles.

My take on it (just my personal reaction) is that it is overblown----the estimates of dark energy and other good cosmological stuff depend on the higher order poles----the smaller bumps that make the CMB skymap all speckly. Cosmologists estimates about the universe don't depend on those few low-order huge bumps. they might be affected by some dust or crud or other unexpected local effect and one would then just factor that out and throw that away and one would still have the essential WMAP picture of the microwave background and all that can be inferred from it.

so if there is something in that coincidence then it just tells us some minor new detail about the solar system that we didnt know, and doesn't affect the picture of the universe at large (guess).

 

[marcus]

we should try to maintain our focus on quantum gravity in this thread.

quantum gravity means quantum models of the universe
specifically of the GEOMETRY of the universe

the 1915 Gen Rel insight was that gravity equals geometry
ultimately you cannot have a quantum theory of gravity unless you
have a model of the evolving geometry of the universe

oddly rather few approaches to quantum gravity
actually model the universe
or even offer a quantum version of Einstein's 1915 Gen Rel equation
(which describes how the geometry of space evolves, along various possible paths called spacetimes)

some of the most visible lines of theory do not bother to quantize the main equation and do not come up with a model of evolving geometry.

so given this anomalous situation and I want to focus on an approach that DOES do the requisite stuff, namely Lorentzian DT.

In DT ("dynamical triangulations") you have a spacetime manifold M and you have a bigspace of all the possible geometries on M, called Geom(M).
And for these researchers Geom(M) is not just an abstract idea but they are able to get a handle on the possible geometries and express them
and code them as data structures into a computer
and RUN the little mothers
and do various kinds of counts and extract statistics on them and get specs.

so this is a hard-edge hands-on approach to quantizing the evolving geometry of the universe

and the bizarre thing is even tho it looks like an obvious thing to do the main bulk of theorists work on stuff with no connection or relevance to it

 

[marcus]

I want to recall a quote from the most recent DT paper

"The idea to construct a quantum theory of gravity by using Causal Dynamical Triangulations was motivated by the desire to formulate a quantum gravity theory with the correct Lorentzian signature and causal properties [14], and to have a path integral formulation which may be closely related to attempts to quantize the theory canonically...."

this is page 3 of http://arxiv.org/hep-th/0411152
Semiclassical Universe from First Principles
by Ambjorn, Jurkiewicz and Loll

it begins a section called "Observing the bounce"

this points up some unresolved issues for me.
one of the two main aim of developing Lorentzian DT, the authors say, is to get close to Loop Quantum Gravity (the canonical approach to quantizing Gen Rel)

and yet there are conspicuous differences
OF COURSE THE BIG BANG SINGULARITY GOES AWAY IN EITHER CASE
but with Loop it is replaced by a real bounce, where a (possibly very small) contracting phase turns inside out to become an expanding universe.

In DT there is also no singularity but in this case the "bounce" seems to get started from nothing-----you don't see a prior contraction.

the cosmological constant appears naturally in Lorentzian DT and must be positive every computer run they do they choose a value for Lambda . I do not understand this. for them it seems related to volume of their model universe.

another conspicuous difference is that in DT I do not see area and volume observables, and I don't see any indication that they will turn out to have discrete values (when and if constructed)

with DT I see a real direct relation to matter and field theory
because DT already has a kind of lattice that QFT is often defined on.
the only real novelty is that it is not a fixed, pre-arranged, lattice, but instead an evolving one.

Anyway, what is puzzling me right now is that there are important differences from LQG, even though the aim is to get a workable path integral approach that connects with LQG

 

[nightcleaner]

this is page 3 of http://arxiv.org/hep-th/0411152
Semiclassical Universe from First Principles
by Ambjorn, Jurkiewicz and Loll

it begins a section called "Observing the bounce"

Hi Marcus

I have been thinking about the bounce and was wondering if my insight could be correct, or if it is contradicted by the math. My idea is that the bounce is not really a bounce at all, in the same sense that the event horizon is not really a membrane. As I recall an observer falling through an event horizon doesn't really see any "there" there. No observable thermal barrior or gravitational tidal effect is measurable by the observer because of distortions in spacetime. Basically, the horizon effect is not observed by a free falling observer because all the gauges are distorted along with the spacetime distortion.

Anyway the horizon which appears to be present in embedding diagrams and in the universe as seen by an outside observer is never actually reached by the free-falling observer, just as you can see the Earth's horizon quite easily but if you set out to find the horizon you discover that you never acturally get there...because there isn't any "there" there, or maybe more precisely, the there that is there is everywhere, so that even when you go there, you find the there that is there is the same as the there that was there before you went there. Ha!

So if the changeover from the approach to infinity to the approach to unity is a horizon, then there will be no locally observable effect on crossover, no bounce. Bounce, after all, implies a change in acceleration, or delta L over T^3. The observer doesn't feel the third factor in the inverse of time because that dimension of time goes to zero as the observer passes through the limit. I am not sure what to call the limit in LQG. It seems to me that it is quite symmetrical, so we could speak of a concave limit and a convex limit, we could speak of a limit approaching infinity (never gets there because of restrictions on the maxumum value of c, which is L/T) and a limit approaching unity (never gets there because of discrete quantum intervals). These limits are different only to the observer watching the fall, and cannot be determined or felt by the falling observer.

Correct me if I am missing something.

Meanwhile, you seem to have a better grasp of the math than I have. Would it be possible for you to translate into words the terms in equation one from your reference, the partition function for Quantum gravity? I mean a literal translation of the formula into English. If it would not be too much trouble for you to do so, I am sure it would help me understand, and perhaps help others also.

I sat in one of Brian Greene's topology classes and was very excited to find that I could understand the formulas he was writing on the board, because as he wrote them he also spoke the meaning of the symbols. For some reason this made an incredible difference to me, and gave me renewed confidence that i could understand the math if only someone would walk me through it a few times. I have been longing desperately for a return to that fluidity of understanding.

When you speak of dynamics in the spacetime structure, as if spacetime itself changes over time in a manner of a history, or of a path integral, how many dimensions are you counting? I guess i could ask, what order equation? For example, velocity is length divided by time, L/T, a first order equation, but acceleration is length over time squared, L/T^2, a second order equation. Spacetime in the Minkowski-Einstein sense is four dimensions of spacetime equivalence, in which case L^3/T, a third order equation? So if we take that as changing over time, are we talking of a mathematical expression involving something like L^3/T^2?

I am feeling confused by this and hope you can help me get clear.

Thanks,

nc

484

 

[Kea]

QG Partition Function

Would it be possible for you to translate into words the terms in equation one from your reference, the partition function for Quantum gravity? I mean a literal translation of the formula into English. If it would not be too much trouble for you to do so, I am sure it would help me understand, and perhaps help others also.

When you speak of dynamics in the spacetime structure, as if spacetime itself changes over time in a manner of a history, or of a path integral, how many dimensions are you counting?

Richard

Your physical intuition is much better than many people I know with
the mathematical expertise.

The first equation in

http://arxiv.org/abs/hep-th/0411152

is $$Z ( \Lambda , G ) = \int \mathcal{D} [g] e^{i S [g]} \hspace{4mm} S[g] = \frac{1}{G} \int \textrm{dx}^{4} \sqrt{| \textrm{det} g |} (R - 2 \Lambda)$$

$\Lambda$ is the cosmological constant. $G$
is Newton's constant. The fancy $\mathcal{D} [g]$ means
that the integral is over some ridiculously huge space of fields; in
this case the metrics $g$.

The $e^{i S [g]}$ is the 'weight' for the contribution
from any given field. $S$ is the action functional, an
integral over a four dimensional manifold. The dimension could be
altered.

Under the square root
is the determinant of $g$, which recall at a given point
is like a matrix. $R$ is the Ricci scalar, defined by
$$R = g^{\mu \nu} R_{\mu \nu}$$ where one uses the Einstein
convention that one sums over matching indices, which range from
1 to 4. Here $R_{\mu \nu}$ is the Ricci tensor, defined by
$$R_{\mu \nu} = R^{\lambda}_{\mu \lambda \nu}$$ in terms
of the Riemann tensor
$$R^{\rho}_{\sigma \mu \nu} = \partial_{\mu} \Gamma^{\rho}_{\nu \sigma} - \partial_{\nu} \Gamma^{\rho}_{\mu \sigma} + \Gamma^{\rho}_{\mu \lambda} \Gamma^{\lambda}_{\nu \sigma} - \Gamma^{\rho}_{\nu \lambda} \Gamma^{\lambda}_{\mu \sigma}$$
using the Christoffel symbol
$$\Gamma^{\lambda}_{\mu \nu} = \frac{1}{2} g^{\lambda \sigma} ( \partial_{\mu} g_{\nu \sigma} + \partial_{\nu} g_{\sigma \mu} - \partial_{\sigma} g_{\mu \nu})$$

Hope this is a little helpful. As you will see, this mathematics falls far short
from being a satisfactory description of 'cosmic bounces', which is
a term combining the 'bounce' of CT (not a great idea as you say)
and the 'cosmic duality' associated to Strings or Cosmic Galois groups
and other fun things.

Regards
Kea

 

[nightcleaner]

Hi Kea. Thank you for the kind words.

So we might say something like "The partition function for quantum gravity in a universe with a given cosmological constant and gravitational constant $$\zeta_(\lamda G)$$ is the integral over the metrics (g) multiplied by the weight factor expressed as the natural log of the action functional for any given field, where the action functional is the inverse of the gravitational constant times the integral in four dimensions of the square root of the absolute value of the determinants of (g) times the ...ok I'm lost.

Is that gamma the inverse of the hyperbolic tangent function? I ran into it once before and came out with the right table of numbers after some coaching.

Anyway I am grateful for your exposition and am still working on it. Just to let you know I am here paying attention.

thanks,

nc

573

 

[Kea]

nc

573

What are the numbers?

$\Gamma$ hasn't anything to do with such functions. It
is the Christoffel symbol associated to a connection, which we need
to understand how to move about the manifold, and to see the
'curvature'.

Try this webite:
Carroll's lecture notes on General Relativity

http://pancake.uchicago.edu/~carroll/notes/

Kea

 

[nightcleaner]

The numbers are pure vanity. Silly, really, but that is how many views there were when i posted. So now there are 608, and i know that there are maybe as many as thirty people checking in.

I will check out the Christoffel symbol. Thanks.

Later... I did check out the link, but my browser didn't want to read that file type. Anyway I looked it up in Wikipedia and amazing coincidence, it is the same topic Brian Greene lectured on in his topology class the day I was there. And it might even be the same gamma I remembered from my work with DW on the relativity board. I noticed on Wiki, although I have not had time to study it, that gamma deals with tangential vectors. Tanh would just be the same vector under relitivity, I guess.

nc

 

[Kea]

You're quite right about tanh, of course. Are you sure it wasn't a
course in Differential Geometry?

 

[Lacy33]

Are you sure it wasn't a
course in Differential Geometry?

No it was an undergrad topology class.

 

[Kea]

Shoshana

No it was an undergrad topology class.

You're lucky. I never had a proper topology course as an
undergrad.

So - do you know Nightcleaner?

 

[Lacy33]

You're lucky. I never had a proper topology course as an
undergrad.

So - do you know Nightcleaner?

Correct. It is difficult to find a topology course at many Universities, but Columbia University offers undergrad topology. My first topology professor at Columbia University was Michael Thaddeus. Amazing speaker!

Yes I am of good fortune to know Richard. He is very fine and uniquely talented.

 

[nightcleaner]

Correct. It is difficult to find a topology course at many Universities, but Columbia University offers undergrad topology. My first topology professor at Columbia University was Michael Thaddeus. Amazing speaker!

Yes I am of good fortune to know Richard. He is very fine and uniquely talented.

Aw, shucks.

Anyway it was only one class, not the whole semester.

I am concerned that Marcus may be mad at us for hijacking his two worlds thread. So I am starting a new thread called hypervisions. Invite all to join in there. Sorry Marcus. hope you are not too angry at me. i am still trying to follow your posts, don't give up on me yet, ok?

nc

 

[Lacy33]

Aw, shucks.

Anyway it was only one class, not the whole semester.

Some people gain a whole degree... ner do anything with it.
And another can attend one class and produce amazing advances.

"A word to the wise is sufficient"

 

[Lacy33]

I am concerned that Marcus may be mad at us for hijacking his two worlds thread.
nc

Are you sure Marcus is not interested in helping you on this thread?
I never saw anyone so devoted to lending a hand and sharing expertise as Marcus has been to you.

Many of us have been watching the two of you and hoping for some wonderful sharing to reveal new things?

 

[nightcleaner]

Thank you Shoshana, and I am grateful to you, and to Marcus, and to others, for helping my understanding.

nc

 

[marcus]

...
I am concerned that Marcus may be mad at us for hijacking his two worlds thread. So I am starting a new thread called hypervisions. Invite all to join in there. Sorry Marcus. hope you are not too angry at me. i am still trying to follow your posts, don't give up on me yet, ok?
...

hi Cleaner,
I didnt experience any vexation on that (or any other) account
rather the opposite (you and Sho and Kea socializing seems appropriate
in a thread partly devoted to combinatorics)

but I have been busy with another bit project away from PF
and haven't felt an urge to join in discussion

I think starting "hypervisions" thread was a good idea----sharing ways of visualizing stuff in more than 3D----I hope you get some wider response.

BTW we might be in a lull until after the 1st of the year.

 

[nightcleaner]

Thanks Marcus. Happy New Year. nc

 

[marcus]

this was a congenial thread (with selfAdj, Richard the nc, Kea, Shoshana, and others occasionally taking part) from around December 04.

I want to keep tabs on it, and perhaps add to it, because of the discussion of CDT (causal dynam. triang.) Monte Carlo moves
following this AJL paper
http://arxiv.org/hep-th/0105267

a new AJL paper just came out, a short one about the running of the spacetime dimension---a surprising result that they discovered in the course of simulating the evolution of many universes on their computer.

in the new AJL shortie they tell us to expect a long paper dated May 05 which will give details on their computer simulations, how they generate random spacetimes, random worlds.

the basic thing is to put a quarter of a million simplexes into the computer and have them be glued together randomly and then have the whole spacetime shebang be shuffled repeatedly by "Monte Carlo moves". there are some halfdozen or so of those simple moves which are adequate, if repeated enough, to explore all possible spacetime histories.

this new May 05 paper will be called "Reconstructing the Universe" and it will probably do a lot of the same things as http://arxiv.org/hep-th/0105267
except it will be more up-to-date. It will probably also show pictures of the various Monty moves and describe step by step how a simulation works.

An interesting historical note is that people (AJL and others) have been trying to make a simplicial model of the 4D universe, and do this kind of computer simulation, for something like 15 years without it working!

Jan Ambjorn has been especially stubborn and there is a long history of partial successes in 2D and 3D, and of repeated failures to get 4D.
then last year, in 2004, it must have felt, for Ambjorn, like someone finally stops beating against a wall and the wall vanishes.
They wrote a paper around April 2004 called "Emergence of a 4D world"

You can get all the links to earlier papers from the biblio in their current paper, should you wish. So I will just give the current one:
http://arxiv.org/hep-th/0505113

 

[marcus]

Getting back to a question asked in post #1 of this thread,

...I would like to know why the research output in DT is essentially flat. Given its apparent promise and the recent (2004) success, why arent more people getting into DT?

just to review. CDT seems to have the required low-energy limit and it has a hamiltonian that they construct explicitly in, for example, that 2001 paper I'm always citing.
and you can do calculations, which you cannot easily do in several other approaches to QG.
(calculations galore, done by the barbaric means of Monte Carlo, which I regard as having the proper empirical spirit.)

so it's got
1. classical limit
2. an interesting (Hartle Hawking wavefunction) semiclassical limit
3. dynamics
4. opportunities to calculate and do computer simulations

but look, while LQG is having 100 some papers a year and has a vigorous kind of quadratic-looking growth curve, the output in CDT is esssentially flat. This roughly tracks the output since 1998 when CDT was invented or began as a research line:

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/1998/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/1999/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2000/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2001/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2002/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2003/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2004/0/1

Last 12 months (abbreviated LTM):
http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/past/0/1

1998   3
1999   3
2000   5 
2001   4
2002   6
2003   4
2004   4
LTM    4

 

[selfAdjoint]

There may be a perception in the community that using fixed triangulations is "clunky" compared to networks and spin foams. And we must never discount the effect of John Baez's advocacy over the years!

But these AJL papers will be sure to change things. As one string physicst said back when the mirror dualities were discovered, maybe even some grad students will become interested!

 

[marcus]

... we must never discount the effect of John Baez's advocacy over the years!
...

actually it was John Baez's advocacy in FAVOR of CDT last year after the Marseille conference in May that galvanized me and may have made CDT a lot more visible to others as well. so that can work both ways


the analogy with a Feynman path integral is very strong
as if instead of one fixed jagged piecewise linear path that a particle might take, one imagines instead the continuum limit of a whole
"blur" or billions of possible zigzag paths

so likewise with CDT one may imagine a glittering blur of millions of triangulations
(not one fixed triangulation)
that averages out to a smooth spacetime shape

 

[marcus]

there is a common element in the last 3 CDT papers. since they carefully repeat it each time maybe we should listen extra carefully:

---from hep-th/0404---

Causal dynamical triangulations are a framework for defining quantum gravity nonperturbatively as the continuum limit of a well-defined regularized sum over geometries. Interestingly, and in complete agreement with current observational data is the fact that the physical cosmological constant $$\Lambda$$ in dynamical triangulations is necessarily positive...

---from hep-th/0411---

Causal Dynamical Triangulations constitute a framework for defining quantum gravity nonperturbatively as the continuum limit of a well-defined regularized sum over geometries. We reported recently on the outcome of the first Monte Car..."

---from hep-th/0505---

"In the CDT approach, quantum gravity is defined as the continuum limit of a regularized version of the nonperturbative gravitational path integral . The set of spacetime geometries to be summed over is represented by a class of causal four-dimensional piecewise flat manifolds (“triangulations”). Every member T of the ensemble of simplicial spacetimes can be wick-rotated to a unique Euclidean piecewise flat geometry, whereupon the path integral assumes the form of a partition function ...

All geometries share a global, discrete version of proper time. In the continuum limit, the CDT time $\tau$ becomes proportional to the cosmological proper time of a conventional minisuperspace model..."

 

[marcus]

... compared to networks and spin foams...

you prompted me to compare CDT with spin foams and it occurred to me that they look rather alike, there is a (superficial?) visual resemblance

but CDT is less dependent, I believe, on coordinates
I think of Georg Riemann around 1850 offering a way to do geometry without prior commitment to a particular metric---but only with some coordinate patches: a shapeless "manifold"---Riemann had to have coordinates.

And then around 1950 came Tulio Regge offering a way to do "general relativity without coordinates" (I think that was his paper's title or something like it).

so then you did not even need to make a prior commitment to coordinate patches (and the DIMENSION always sneaks in as the number of coordinate functions, the "n" in the local resemblance to Rⁿ.)

So CDT as a Regge offspring makes less prior commitments. It doesn't have an essential need for coordinates because the "geometry is all in the gluing". the geometry is all in how the many identical cells are stuck together.

and although one chooses at the outset cells of a particular dimension (for instance simplices which are pieces of Minkowski space) when they are glued together it turns out that the dimension can be determined in some empirical ways (e.g. by running diffusion) and it does not need to come out exactly 4, or even to be an integer. it is, after all, not a DIFFERENTIABLE manifold that one makes by gluing the cells together. this seems very strange to me but it does look like one is going further in the direction of not having prior commitments to anything.

and yet they are able to define a transfer matrix and a hamiltonian and to calculate and to do computer simulations, which a lot of people working with stuff on differentiable manifolds do NOT seem able to do. so it is paradoxical----they throw out---they make less prior commitment to structure---and yet they seem able to calculate MORE rather than less----like they even get statistics about their spacetimes----having made so many of them that they must record statistics.

utrecht interests me, now 't Hooft and Robbert Dijkgraaf, and Renate Loll, and Jan Ambjorn are there and the last 3 of them are invited speakers at the October "Loops 05" conference. I am thinking that Utrecht work will be fairly noticeable at this year's conference

 

[selfAdjoint]

It's an interesting and I believe deep toplogical question how much you actually predetermine the coordinate stuff with this "gluing"? There was always this uneasy feeling that AJL had presetermined pseudo-Riemannian with their "causality" - see also causal sets which seems to do that without the gluing.

Back in the grad school day, I was always impressed by the fact that the left side of the Riemann-Roch index theorem was pure toplogy and the right side was analytic. Of course since then we have the same thing much bigger with Atiyah-Singer. And look at the run BRST has had! Save your Jordan curves boys, topology will rise again!

 

[marcus]

It's an interesting and I believe deep toplogical question how much you actually predetermine the coordinate stuff with this "gluing"? There was always this uneasy feeling that AJL had presetermined pseudo-Riemannian with their "causality" - see also causal sets which seems to do that without the gluing.

Back in the grad school day, I was always impressed by the fact that the left side of the Riemann-Roch index theorem was pure toplogy and the right side was analytic. Of course since then we have the same thing much bigger with Atiyah-Singer. And look at the run BRST has had! Save your Jordan curves boys, topology will rise again!

I'll talk around the issues you raise here, which are interesting ones, though unable to address them head-on. Another way to think of the gluing of 4-simplices together is as carving up the S³ x R carcass.

AJL start by putting a topological S³ x R into the computer.

for technical reasons they specify that there are going to be, say, 80 time steps and some total number of simplexes, like 200,000.
this S³ x R gets repeatedly divided up into around 200,000 4simplexes, and they takes their statistical measures of what results.
and then they start again, using a different total number, like 300,000

their idea is that the CDT theory is the continuum limit

they say spacetime is not discrete (or that they have seen no indication of a fundamental discreteness, in that sense they are at odds with LQG which has seen a discrete spectrum of area and volume operators)

they say they have seen no indication of a minimum length

their spacetime, their universe, is a continuum limit of quantum geometries that are established in a way that does not use coordinates or metrics or connections, but only uses simplexes (instead of coordinates)

but they do not think the world is made of simplexes, that is just what they use to describe a sequence of finer and finer geometries.

now I have to think how to relate what I've just said to the issues you raised

 

[marcus]

I'll talk around the issues you raise here, which are interesting ones, though unable to address them head-on. Another way to think of the gluing of 4-simplices together is as carving up the S³ x R carcass.

AJL start by putting a topological S³ x R into the computer.
...

they put S³ x R into the computer and carve it up, in the case where they are using 4-simplexes as the basic relational cell (I think of the 4-simplex, which is a tiny piece of minkowski space, as an ATOM OF RELATIONSHIP, or as an atom of spacetime relation and causality)

in the cases where they have used 2-simplexes or 3-simplexes as the basic relational buildingblock, then they used a different topological carcass to carve up. they used S¹ x R, or else S² x R.

now the interesting thing, or one very interesting, is that if you don't use the right carving rules (dont make it causal or Lorentzian but only euclidean) then you can get any dimension. Like with S² x R
dividing up into 3-simplexes it does not have to come out 3D. You can get a beast that is 2D or that is INFINITE dimensional. unbounded (infinite in the limit) anyway.
the infinite dimensionality comes from too much connectivity

like think of the Earth surface with the N hemisphere carved into 360 skinny triangles each one with its vertex at the N pole. this is too connected because every spot is in the same triangle as the N pole. One has to have a carving rule that makes this kind of thing unlikely.

the point I am making is that DIMENSIONALITY MUST EMERGE and it is not predetermined by the fact that you started with a topological space that is
S³ x R
(because a mere topological space does not have preordained dimension, and when it gets some structure that permits dimension to be defined then that dimension does not have to be the same at every point)
and the dimensionality that emerges is also not predetermined by the fact that we happen to be using 4-simplexes to carve the S³ x R up in.

Because the topological space can be partitioned into 4-simplexes in such a way as to give very high dimension or very low dimension to various locations in it (the S³ x R carcass doesn't have any geometry until the 4-simplexes come and partition it)

 

[marcus]

I had better quote AJL about this, because it is a key point, and I will repost the basic CDT reading list in case anyone wants to refer to the original papers

This is from the introductory paragraph of "Emergence of a 4D World", reference 2 below:

"... a particular case of the more general truth, not always appreciated, that in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry. (By dimension we mean an effective dimension observed at macroscopic scales.)..."

Dimension is just another aspect of geometry. AJL are working with a topological continuum which has no prior geometry and they put the geometry on in a quantum way, and dimensionality emerges, and it is different at different scales, and down near Planck scale it tends to be around D = 2, or even less. But happily enough at big scale, where we like to build our houses and go snow-boarding etc, it is right around D = 4.

Here is the short list of CDT references

1.
http://arxiv.org/hep-th/0105267
Dynamically Triangulating Lorentzian Quantum Gravity
J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)
41 pages, 14 figures
Nucl.Phys. B610 (2001) 347-382
"Fruitful ideas on how to quantize gravity are few and far between. In this paper, we give a complete description of a recently introduced non-perturbative gravitational path integral whose continuum limit has already been investigated extensively in d less than 4, with promising results. It is based on a simplicial regularization of Lorentzian space-times and, most importantly, possesses a well-defined, non-perturbative Wick rotation. We present a detailed analysis of the geometric and mathematical properties of the discretized model in d=3,4..."

2.
http://arxiv.org/abs/hep-th/0404156
Emergence of a 4D World from Causal Quantum Gravity
J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3) ((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian University, Krakow, (3) Spinoza Institute, Utrecht)
11 pages, 3 figures; final version to appear in Phys. Rev. Lett
Phys.Rev.Lett. 93 (2004) 131301
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over geometries in nonperturbative quantum gravity, with a positive cosmological constant. We present evidence that a macroscopic four-dimensional world emerges from this theory dynamically."

3.
http://arxiv.org/abs/hep-th/0411152
Semiclassical Universe from First Principles
J. Ambjorn, J. Jurkiewicz, R. Loll
15 pages, 4 figures
Phys.Lett. B607 (2005) 205-213
"Causal Dynamical Triangulations in four dimensions provide a background-independent definition of the sum over space-time geometries in nonperturbative quantum gravity. We show that the macroscopic four-dimensional world which emerges in the Euclidean sector of this theory is a bounce which satisfies a semiclassical equation. After integrating out all degrees of freedom except for a global scale factor, we obtain the ground state wave function of the universe as a function of this scale factor."

4.
http://arxiv.org/abs/hep-th/0505113
Spectral Dimension of the Universe
J. Ambjorn (NBI Copenhagen and U. Utrecht), J. Jurkiewicz (U. Krakow), R. Loll (U. Utrecht)
10 pages, 1 figure
SPIN-05/05, ITP-UU-05/07

"We measure the spectral dimension of universes emerging from nonperturbative quantum gravity, defined through state sums of causal triangulated geometries. While four-dimensional on large scales, the quantum universe appears two-dimensional at short distances. We conclude that quantum gravity may be "self-renormalizing" at the Planck scale, by virtue of a mechanism of dynamical dimensional reduction."

 

[marcus]

...This is from the introductory paragraph of "Emergence of a 4D World", reference 2 below:

"... a particular case of the more general truth, not always appreciated, that in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry. (By dimension we mean an effective dimension observed at macroscopic scales.)..."

Dimension is just another aspect of geometry. AJL are working with a topological continuum which has no prior geometry and they put the geometry on in a quantum way, and dimensionality emerges, and it is different at different scales, and down near Planck scale it tends to be around D = 2, or even less. But happily enough at big scale, where we like to build our houses and go snow-boarding etc, it is right around D = 4.

Here is the short list of CDT references...

Now #5 on the short list has appeared

5.
http://arxiv.org/hep-th/0505154
Reconstructing the Universe

and we have another new title in the "in preparation" or "to appear" category, which is the first CDT paper AFAIK about BLACK HOLES.

This is to be a collaboration of Renate Loll with BIANCA DITTRICH.

The two of them co-authored a CDT paper back in 2002 but since then Dittrich has been doing stuff with Thiemann "Master Constraint" program and also with partial-observable, relational-time. In effect she has been helping Thiemann rake his chestnuts out of the fire.

Dittrich work on Master Constraint has been major. She is an interesting case because she is both active currently in LQG/LQC and also has a significant interest in CDT.

Also it seems like a high priority job to see how CDT deals with black holes
(we already know how CDT does the big bang non-singularity, or seem to know, all the AJL simulations portray that, but what AJL show is a vacuum state universe so far with no black holes)

the Dittrich-Loll paper to appear is called
Counting a Black Hole in Lorentzian Product Triangulations

It would not surprise me if Dittrich moved from AEI-Potsdam to Utrecht. The Utrecht group seems to me to be growing.

[EDIT in case anyone is curious the 2002 paper of Dittrich-Loll was
http://arxiv.org/abs/hep-th/0204210 ]

 

[marcus]

Basic incompatibility between CDT and LQG (removable?)

the basic incompatibility is that LQG is based on a smooth continuum and CDT on an unsmooth. CDT is the limit of piecewise flat manifolds, like a Feynman path is a superposition of piecewise linear paths and the result is not differentiable

neither CDT nor LQG is DISCRETE. but in LQG the underlying space is a differentiable manifold (smooth although without prior metric)
and in CDT the underlying space is not smoothly coordinatized, it is NOT a differentiable manifold. it is all jig-jaggy piecewise flat. and it does not have any coordinates.

in 1850 Riemann invented how to do a continuum without prior metric (but with smooth functions as coordinates, the point is the patches)

in 1950 Tulio Regge invented how to do a continuum and write Einstein Gen Rel on it WITHOUT COORDINATES, no differentiable functions.

So this is a major disconnect---it looks as if Renate Loll is A HUNDRED YEARS MORE ADVANCED, in the history of mathematics, than, say Lee Smolin and Carlo Rovelli. this could be serious.

But maybe it can be fixed. Maybe one can do LQG on a series of finer and finer piecewise flat manifolds. simplicial manifolds. whatever.
(here "flat" means 4D minkowski, the usual special relativity 4D flat)

the thing that Renate Loll can say (and the Utrecht gang have their knives out and mean business these days) is what I quoted a post back:
in any nonperturbative theory of quantum gravity dimension will become a dynamical quantity, along with other aspects of geometry.

Because LQG is built on a smooth manifold of some chosen dimension, with exactly 3 coordinate functions, or exactly 4, it is so-to-say stuck with that choice of dimension. But a piecewise flat manifold can be so wrinkled and kinky and crumpled and frazzled that it loses track of what its dimension is.
at least at small scale and maybe the real world is like that

this gives the Utrecht people a strong card to play

 

[marcus]

Now #5 on the short list has appeared

5.
http://arxiv.org/hep-th/0505154
Reconstructing the Universe

...

Here is the first paragraph of Reconstructing the Universe:

"Nonperturbative quantum gravity can be defined as the quest for uncovering the true dynamical degrees of freedom of spacetime geometry at the very shortest scales. Because of the enormous quantum fluctuations predicted by the uncertainty relations, geometry near the Planck scale will be extremely rugged and nonclassical. Although different approaches to quantizing gravity do not agree on the precise nature of these fundamental excitations, or on how they can be determined, most of the popular formulations agree that they are neither the smooth metrics g_mu,nu (or equivalent classical field variables) of general relativity nor straightforward quantum analogues thereof. In such scenarios, one expects the metric to re-emerge as an appropriate description of spacetime geometry only at larger scales."

Renate Loll can write English better than most of us nativespeakers, or else it is an AJL team phenomenon. Anyway the style of the AJL writings is usually strong and clear. This helps (them at least).

Now that String seems less promising, seems off track in fact, the other approaches to Quantum Gravity which have so far been mostly at peace under the big Loop tent, are becoming more disposed to controversy among themselves.

Here right away at the beginning, the AJL writer is playing the strong card against LQG. The suggestion is that LQG cannot be right because it is too smooth at small scale. How can LQG be right? since it is built on a smooth differentiable manifold of fixed dimension, and uses "classical field variables" equivalent to the metric, namely the connections. Actually LQG has made a lot of progress and will have accomplished a lot even if it eventually turns out to have been a PILOT STUDY for something else. It has tackled the big bang and inflation and black holes and produced a lot of new ideas like Freidel/Starodubtsev and for instance lately Perez/Rovelli about the Immirzi parameter. Probably there is some stuff it DOES get right. Thiemann has started confirming details of Bojowald's LQC big bang using the full LQG theory etc. Plenty is happening in LQG. But the AJL writer hints how can LQG be fundamentally correct? since "because of the enormous quantum fluctuations predicted by the uncertainty relations, geometry near the Planck scale will be extremely rugged and nonclassical."

Well maybe that is not as aggressive as it sounded to me. Maybe everybody inside the big Loop tent feels that he or she is enough rugged and nonclassical. Maybe spin networks are that. So one can hear this opening paragraph different ways.

There may be some fire at the Potsdam conference in October.
I think String has dwindled so that there is less of a common rival and less of a collective spirit. So people and ideas can wrangle with each other more openly, which I guess is traditional in science so must be OK

 

[selfAdjoint]

I'll talk around the issues you raise here, which are interesting ones, though unable to address them head-on. Another way to think of the gluing of 4-simplices together is as carving up the S³ x R carcass.

AJL start by putting a topological S³ x R into the computer.

for technical reasons they specify that there are going to be, say, 80 time steps and some total number of simplexes, like 200,000.
this S³ x R gets repeatedly divided up into around 200,000 4simplexes, and they takes their statistical measures of what results.
and then they start again, using a different total number, like 300,000

their idea is that the CDT theory is the continuum limit

they say spacetime is not discrete (or that they have seen no indication of a fundamental discreteness, in that sense they are at odds with LQG which has seen a discrete spectrum of area and volume operators)

they say they have seen no indication of a minimum length

their spacetime, their universe, is a continuum limit of quantum geometries that are established in a way that does not use coordinates or metrics or connections, but only uses simplexes (instead of coordinates)

but they do not think the world is made of simplexes, that is just what they use to describe a sequence of finer and finer geometries.

now I have to think how to relate what I've just said to the issues you raised

Two responses to this:

I. Maybe CDT will wind up doing with LQG what the lattice does with the standard model: achieve geniune if spotty non-perturbative results while the lion's share of the physics is being done n-loop perturbative with the continuum theory. (n being a small integer).

II. Topology has lots of more general things than polyhedra (which is what triangulated manifolds are). CW-complexes, ANRs, lots of things in graduated sequences of generality. And there are all those lovely categories.